UK Nonlinear News, March 2003
Hardback, £72.00 (Amazon)
Dynamical systems and control theory are two subjects that ought to have a great deal in common. Yet, somehow, the two communities seem to have evolved largely independently, and often speak somewhat different languages. As a result, there is far less interaction between the two fields than there ought to be. In my opinion, this is a great loss to both sides, but despite the efforts of various committed individuals, the barriers between the two areas are proving remarkably difficult to break down. The key issue is perhaps that in control theory, one has to consider arbitrary time-varying inputs to a system, whilst nonlinear dynamics has concentrated on either autonomous systems, or periodically driven ones (which can be treated as autonomous through the usual device of regarding time as an extra state space variable).
The present volume is an impressive, and intriguing, attempt to develop an integrated account of the mathematical connections between nonlinear control, dynamical systems and time-varying perturbed systems. The key unifying concept is to regard all the possible time-varying inputs as a shift space (usually infinite dimensional) driving the dynamical system of interest. In this way a driven system becomes a so called skew product over the driving shift space and can now be regarded as an autonomous system. In essence, this is the same principle as that used to convert periodically driven systems into autonomous ones, except that instead of enlarging the system by a single variable, we typically add an infinite number. Nevertheless, this device allows the use of many standard techniques and results from nonlinear dynamics. Such an idea is also central to some modern approaches to stochastic dynamics, and in particular to Ludwig Arnoldís seminal work on Random Dynamical Systems (reviewed in UK Nonlinear News 17, August 1999). This commonality is no coincidence, since Arnold was Wolfgang Kliemannís PhD supervisor, and Fritz Colonius also studied at Bremen. In some ways the present volume can be seen as a continuation of Arnoldís programme, applied specifically to control systems. The main difference in the present volume is that more regularity is assumed, so that the resulting skew product is continuous. This allows the authors to apply a variety of concepts from topological dynamics and ergodic theory. The exciting aspect of this is that there turn out to be intimate connections between control theory properties of the control system, and the dynamical properties of the skew product. Thus for instance there is a one-one correspondence between control sets and topologically mixing invariant sets for the skew product.
The end result is a treatment which is highly accessible to someone familiar with the basic concepts of dynamical systems. At the same time, the particular applications will be largely unknown, and hence the book provides a wealth of new ideas to explore. I would therefore recommend almost anyone with a background in nonlinear dynamics to read it for this reason alone. Having no expertise in control theory, I am unable to judge how this volume will be perceived within that community, but again would imagine that it could provide a fresh perspective for those interested in the more theoretical aspects of control.
UK Nonlinear News would like to thank Birkhauser for providing a review copy of this book.
A listing of books reviewed in UK Nonlinear News is available.